Computer tomography method using redundant measured values

ABSTRACT

The invention relates to a computer tomography method in which a radiation source moves relative to an examination region along, in particular, a helical or circular trajectory. Measured values are acquired by a detector unit and a CT image of the examination region is reconstructed from these measured values. In the reconstruction, a complementary measured value, whose ray is oriented parallel to the ray of the respective measured value that has been acquired but in the opposite direction thereto, is determined for each of at least some measured values that lie within a reconstruction window. Redundant measured values are used to calculate the complementary measured values, with the help in particular of John&#39;s equation. The measured values for which complementary measured values have been determined are each replaced by a sum comprised a measured value that has been weighted and a complementary measured value that has been weighted, and a CT image is reconstructed, in particular by an exact method of reconstruction, from the replacement measured values, and where appropriate from acquired measured values, that lie within the reconstruction window.

The invention relates to a computer tomography method in which themeasured values used to reconstruct a CT image are only ones that liewithin a reconstruction window, and it relates in particular to exactcomputer tomography methods. The invention also relates to a computertomograph for performing the said method and to a computer program forcontrolling the computer tomograph.

In known methods of the above-mentioned kind, a radiation source movesalong a helical or circular trajectory relative to an examination regionin which the object to be examined is situated. As it does so, adetector unit acquires measured values from which a CT image, such as,for example, an image of the distribution of absorption in theexamination region, can be reconstructed. Many reconstruction methods,and particularly the exact reconstruction methods, are able to use onlya proportion of the measured values. The proportion of the measuredvalues that can be used for reconstruction may for example be defined bya range of solid angles through which the radiation source moves while apoint in the examination region that is to be reconstructed is beingirradiated, or by a given region on the detector unit. The region inwhich measured values that are used for reconstruction are situated iscalled a reconstruction window. Measured values outside thereconstruction window are ignored by methods of the kind mentioned andare termed redundant measured values. A reconstruction method of thiskind has been described by, for example, Katsevich in “Analysis of anExact Inversion Algorithm for Spiral Cone-Beam CT”, Physics Medicine andBiology, vol. 47, pp. 2583-2597 (E1). In this case, the radiation sourcemoves along a spiral, i.e. helical, trajectory relative to theexamination region and the measured values that are used are only theones that lie within what is termed the PI window, which will beexplained in detail below. Things that are disadvantageous about suchmethods are that they have a poor signal-to-noise ratio and that, if theobject moves, they produce motion artifacts, which are a nuisance.

It is therefore an object of the present invention to specify a computertomography method of the kind stated in the opening paragraph in whichthe signal-to-noise ratio is improved and the motion artifacts arereduced.

This object is achieved in accordance with the invention by a computertomography method having the following steps:

-   a) generation, by a radiation source, of a conical beam of rays that    passes through an examination region and an object situated therein,-   b) production of a relative movement between the radiation source on    the one hand and the examination region on the other hand, which    movement comprises at least a rotary movement about an axis of    rotation and is in particular in the form of a helix or circle,-   c) acquisition, by a detector unit and during the relative movement,    of measured values that depend on the intensity in the beam of rays    on the farther side of the examination region,-   d) determination, with the help of redundant measured values, of a    complementary measured value for each of at least some of the    measured values that were acquired in step c) and that lie within a    reconstruction window, the rays associated with the given measured    value and the complementary measured value belonging to it being    oriented in opposite directions to one another,-   e) replacement of each measured value for which a complementary    measured value was determined in step d) by a sum comprising the    measured value, having been weighted, and the complementary measured    value, having been weighted,-   f) reconstruction of a CT image of the examination region from the    measured values lying within the reconstruction window.

In contrast to known computer tomography methods of the kind defined inthe opening paragraph, redundant measured values are used to determinecomplementary measured values that, when added to their respectiveassociated measured values, replace the latter, which means thatredundant measured values too contribute to the reconstruction thatfollows. The invention is based on the finding that, in computertomography methods in which the reconstruction is not confined to only aproportion of the measured values, motion artifacts are reduced and thesignal-to-noise ratio is improved. Taking account of the redundantmeasured values therefore results not only in an improvement in thesignal-to-noise ratio but also in a reduction in the motion artifacts.

In claim 2, if the complementary measured value has not been acquired instep c) then it is determined by means of John's equation. This producesCT images of particularly good quality.

John's equation is a known partial differential equation that, amongother things, allows further measured values to be determined formeasured values that have been acquired, which further measured valuesare consistent with those that have been acquired.

The term “consistent” can be explained as follows. The object in theexamination region is not clearly defined by the set of measured valuesacquired in step c) because the measured values are affected by noise oralso, for example, because too few measured values have been acquired.For the set of measured values, there are therefore a number of possibleobjects in the examination region that fit the measured values. Acomplementary measured value is consistent with the measured values thathave been acquired when it can be produced by the passage of a raythrough one of the possible objects. A consistency condition with whicha complementary measured value of this kind can be determined is forexample John's equation, which is known.

In claim 3, when the weighted complementary measured values and therespective weighted measured values belonging to them are added, acomplementary measured value that has been obtained is given a heavierweighting if it was acquired in step c) rather than having beendetermined by, for example, John's equation. The greater account that isthereby taken of values that have actually been measured gives images ofan improved quality.

Claims 4 and 5 describe a preferred exact computer tomography methodthat produces CT images of good quality for helical trajectories.

A computer tomograph for performing the method according to theinvention is described in claim 6. Claim 7 defines a computer programfor controlling the computer tomograph claimed in claim 6.

These and other aspects of the invention are apparent from and will beelucidated with reference to the embodiments described hereinafter.

In the drawings:

FIG. 1 is a schematic view of a special embodiment of computer tomographwith which a special embodiment of the method according to the inventioncan be performed.

FIG. 2 is a flow chart of the method according to the invention.

FIG. 3 shows a PI straight line and a PI interval for a point in theexamination region.

FIG. 4 shows a PI straight line and the PI interval for a point in theexamination region, when projected onto a plane perpendicular to theaxis of rotation.

FIG. 5 is a schematic view of a planar detector having a PI window.

FIG. 6 is a schematic view in cross-section of a helical trajectory andof an examination region, showing a direct ray and a complementary ray.

FIG. 7 is a flow chart for an exact reconstruction of a CT image.

FIG. 8 is a schematic perspective view of parallel rays in differentpositions.

FIG. 9 is a schematic perspective view of the helical trajectory showinga κ plane and a κ line.

FIG. 10 is a schematic perspective view of the helical trajectoryshowing, in parallel planes, fans of rays formed by rebinning.

The computer tomograph shown in FIG. 1 comprises a gantry 1 which isable to rotate about an axis of rotation 14 extending parallel to the zdirection of the coordinate system shown in FIG. 1. For this purpose,the gantry 1 is driven by a motor 2 at an angular velocity that ispreferably constant, but adjustable. Fastened to the gantry 1 is aradiation source S such as an X-ray generator, for example. This isprovided with a collimator arrangement 3 that extracts from theradiation produced by the radiation source S a conical beam of rays 4,i.e. a beam of rays that is of a finite extent other than zero both inthe z direction and also in a direction perpendicular thereto (i.e. in aplane perpendicular to the axis of rotation).

The beam of rays 4 passes through a cylindrical examination region 13 inwhich an object, e.g. a patient on a patient presentation table (neitherof which is shown) or even a technical object, may be situated. Havingpassed through the examination region 13, the beam of rays 4 impinges ona detector unit 16 fastened to the gantry 1, which detector unit 16 hasa face that comprises a plurality of detector elements that, in thisembodiment, are arranged in the form of a matrix in rows and columns.The columns of detector elements extend parallel to the axis of rotation14. The rows of detector elements are situated in planes perpendicularto the axis of rotation and, in this embodiment, are situated on an arcabout the radiation source S (thus forming a focus-centered detectorunit). In other embodiments however, they may be differently configured,e.g. to describe an arc about the axis of rotation 14 or in a straightline. In any position of the radiation source, each detector element onwhich the beam of rays 4 impinges gives a measured value for one rayfrom the beam of rays 4.

The included angle of the beam of rays 4, which is designated α_(max),determines the diameter of the object cylinder within which the objectbeing examined is situated when the measured values are being acquired.The included angle is defined in this case as the angle that is madewith a plane defined by the radiation source S and the axis of rotation14 by a ray that, in a plane perpendicular to the axis of rotation 14,is situated at the edge of the beam of rays 4. The examination region13, i.e. the object or the patient presentation table, can be displacedby means of a motor 5 parallel to the axis of rotation 14, i.e. to the zaxis. However, to give an equivalent action, the gantry could equallywell be displaced in this direction. When the object is a technical onerather than a patient, it can be rotated in the course of an examinationwhile the radiation source S and the detector unit 16 remain stationary.

If the motors 2 and 5 run simultaneously, the radiation source S and thedetector unit 16 describe a helical trajectory relative to theexamination region 13. If on the other hand the motor 5 for advance inthe direction of the axis of rotation 14 does not run and the motor 2causes the gentry to rotate, a circular trajectory is produced for theradiation source S and the detector unit 16 relative to the examinationregion 13. In the present embodiment, only the helical trajectory willbe considered. The method according to the invention could, however,also be applied where the trajectory was circular.

The measured values acquired by the detector unit 16 are fed to areconstructing computer 10 that is connected to the detector unit 16 by,for example, a data transmission system (not shown) that operateswithout electrical contacts. The reconstructing computer 10 reconstructsthe CT image and reproduces it, on a monitor 11, for example. The twomotors 2 and 5, the reconstructing computer 10, the radiation source Sand the transfer of the measured values from the detector unit 16 to thereconstructing computer 10 are controlled by a control unit 7.

FIG. 2 shows the flow of a method of measurement and reconstruction thatcan be performed with the computer tomograph shown in FIG. 1.

After initialization in step 101, the gantry rotates at an angularvelocity that is constant in the present embodiment. The velocity mayhowever equally well vary, e.g. as a function of time or of the positionof the radiation source.

In step 103, the examination region, i.e. the object or the patientpresentation table, is displaced parallel to the axis of rotation andthe radiation from the radiation source S is switched on, thus enablingthe detector unit 16 to detect the radiation in a plurality of angularpositions. In the course of this, the radiation source S moves relativeto the examination region along a helical trajectory of which theparameters can be defined by

$\begin{matrix}{{y(\lambda)} = \begin{pmatrix}{R\;\cos\;\lambda} \\{R\;\sin\;\lambda} \\{h\;\lambda}\end{pmatrix}} & (1)\end{matrix}$

Here, 2πh is the feed of the table per rotation, λ is the angularposition of the radiation source relative to a reference angularposition that is any desired angular position but fixed, and R is theradius of the helical trajectory 17.

In the present embodiment, the advance per rotation is selected to besuch that, from any point x in the examination region, the radiationsource S is visible over an angular range of at least 180°. The pickingup of measured values with a table advance of this kind is referred toas PI acquisition

PI acquisition will be explained in detail in what follows. For thispurpose, there are shown in FIG. 3 the helical trajectory 17 along whichthe radiation source moves relative to the point x in the examinationregion, and the section I_(PI)(x) of the helix 17 that is cut off by aPI straight line 37. FIG. 4 shows the helix 17 from FIG. 3 projectedonto a plane oriented perpendicularly to the axis of rotation. The PIstraight line 37 is that line which cuts the helix at two points and thepoint x, with the section I_(PI)(x) of the helix that is cut off by thestraight line extending through an angle of less than 2π. In PIacquisition, rays that start from the radiation source S while thelatter is on section I_(PI)(x) of the helix pass through the point x.

In the context of the present embodiment, what is used below is a methodof reconstruction that employs only measured values whose associatedrays originate from section I_(PI)(x) of the helix, i.e. the measuredvalues used are only ones that lie in what is called the PI interval. Itis known that, by considering the geometry, it can be shown thatmeasured values whose rays start from a given section of the helix liewithin a reconstruction window on the detector unit. In the present PIacquisition, the reconstruction window is the PI window. The PI window25 is bounded on a planar detector 60 by the PI lines 21, 23 that aredefined by the following equations:

$\begin{matrix}{{{v(u)} = {{+ \frac{h}{2\pi}}\left( {1 + \left( \frac{u}{R} \right)^{2}} \right)\left( {\frac{\pi}{2} - {\arctan\;\frac{u}{R}}} \right)}}{and}} & (1) \\{{v(u)} = {{- \frac{h}{2\pi}}\left( {1 + \left( \frac{u}{R} \right)^{2}} \right)\left( {\frac{\pi}{2} + {\arctan\;\frac{u}{R}}} \right)}} & (2)\end{matrix}$

Here, u and v are coordinates on the planar detector 60 in thecoordinate system shown in FIG. 5. For reasons of clarity, thecoordinate system has been shown below the planar detector 60. However,the origin of the coordinate system is situated at the center of thedetector. The unit vectors of the coordinate system are 1_(u)=(−sin λ,cos λ, 0) and 1_(v)=(0, 0, 1).

The planar detector 60 is an imaginary detector that contains the axisof rotation 14 and is oriented perpendicularly to that ray that,starting from the position of the radiation source at the time, strikesthe axis of rotation 14 perpendicularly. Measured values that aredetected by the actual detector unit 16 can be projected onto the planardetector 60 along their corresponding rays.

Computer tomography methods of the kind defined in the opening paragraphuse only measured values that lie within a reconstruction window. Therest of the measured values are not used and this leads to a poorsignal-to-noise ratio and motion artifacts. In the following steps, acomplementary measured value g_(c)(λ₀,u₀,v₀) is determined, for eachmeasured value g(λ₀,u₀,v₀) that lies within the reconstruction window,with the help of redundant data, i.e. measured values that lie outsidethe reconstruction window, as a result of which the signal-to-noiseratio is improved and the motion artifacts reduced. The ray associatedwith the measured value g(λ₀,u₀,v₀) is referred to below as the directray.

In other embodiments, a complementary measured value may be determinedfor each of only a proportion of the acquired measured values that liewithin the reconstruction window. In particular, it would be possiblefor a complementary measured value to be determined only for thosemeasured values that are situated in the edge regions of thereconstruction window.

A complementary measured value is defined by the path followed by thecomplementary ray belonging to it. A complementary ray follows a pathparallel to the direct ray belonging to it but in the opposite directionthereto. A consistency condition can be used to determine thecomplementary measured value associated with the complementary ray. Theconsistency condition that is used in the present embodiment todetermine the complementary measured value associated with thecomplementary ray is the equation generally referred to as John'sequation that was published in “Partial Differential Equations”, F.John, Applied Mathematical Sciences, Springer-Verlag, 1971 and that hasbeen adapted for helical trajectories in U.S. Pat. No. 6,292,526 B1.

In accordance with the invention, the method used to determine acomplementary measured value may be any method, and in particular anyconsistency condition, that, by taking account of redundant measuredvalues, allows a complementary measured value to be determined, at leastapproximately, for a complementary ray that follows a path parallel tothe direct ray but in the opposite direction thereto.

To determine the particular complementary measured valueg_(c)(λ₀,u₀,v₀), in step 105 an approximating measured valueg(λ₁,u₁,{tilde over (v)}₁), whose associated approximating ray 31follows approximately the same path as the complementary ray (see FIG.6), is first determined from the acquired measured values. Thisapproximating ray starts from the angular position λ1 and impinges onthe planar detector 60 at a point (u₁, {tilde over (v)}₁).

The angular position of the approximating ray is defined by

$\begin{matrix}{\lambda_{1} = \left\{ \begin{matrix}{{\lambda_{0} + \pi - {2\beta\mspace{14mu}{where}\mspace{14mu} v_{0}}} > 0} \\{{\lambda_{0} - \pi - {2\beta\mspace{14mu}{where}\mspace{14mu} v_{0}}} < 0}\end{matrix} \right.} & (3)\end{matrix}$

Here, β is the fan angle of the direct ray, i.e. the angle that thedirect ray makes with a ray that intersects the planar detector 60 inFIG. 5 in the center. The angular position λ₁ is the angular position atwhich a direct ray that starts from the position y(λ₀) of the radiationsource and impinges on the planar detector at a point (u₀, v₀), firsttouches the helix 17 if the ray is displaced parallel to the axis ofrotation 14.

The equation that governs the u coordinate of the approximating ray onthe planar detector 60 is:u ₁ =−u ₀   (4)

The v coordinate is determined as follows. By λ=λ₁ and u=u₁ is defined afan of rays that starts from the angular position λ₁ on the helix 17 andimpinges on all those detector elements for which u=u₁ is true. Therespective point of intersection with the direct ray 33 is determinedfor each ray in this fan of rays. The approximating ray 31 is then thatray in the fan of rays whose point of intersection is closest to theaxis of rotation 14. This approximating ray impinges on the planardetector at a point (u₁,{tilde over (v)}₁), where

$\begin{matrix}{{\overset{\sim}{v}}_{1} = \left\{ \begin{matrix}{{{\frac{\sqrt{u_{1}^{2} + R^{2}}}{R}\left\lbrack {{{h\left( {{2\beta} - \pi} \right)}\left( {\frac{1}{2\cos\;\beta} - \frac{\sqrt{u_{1}^{2} + R^{2}}}{R}} \right)} - {2v_{0}\cos\;\beta}} \right\rbrack}\mspace{14mu}{where}\mspace{14mu} v_{0}} > 0} \\{{{\frac{\sqrt{u_{1}^{2} + R^{2}}}{R}\left\lbrack {{{h\left( {{2\beta} + \pi} \right)}\left( {\frac{1}{2\cos\;\beta} - \frac{\sqrt{u_{1}^{2} + R^{2}}}{R}} \right)} - {2v_{0}\cos\;\beta}} \right\rbrack}\mspace{14mu}{where}\mspace{14mu} v_{0}} < 0}\end{matrix} \right.} & (5)\end{matrix}$

In step 107, the path followed by the complementary ray is determined.As has already been explained above, the complementary ray follows apath parallel, but in the opposite direction, to the direct ray 33. Thedirect ray 33 impinges on the surface 35 of the cylinder that is definedby the helical trajectory 17 at a point

$\begin{matrix}{z = {z_{0} = {{h\;\lambda_{0}} + {v_{0}{\frac{2R\;\cos\;\beta}{\sqrt{u_{1}^{2} + R^{2}}}.}}}}} & (6)\end{matrix}$

During the acquisition of the approximating measured value, theradiation source S was situated at a point

$\begin{matrix}{z = {z_{1} = {{h\;\lambda_{1}} = \left\{ \begin{matrix}{{{h\left( {\lambda_{0} + \pi - {2\;\beta}} \right)}\mspace{14mu}{where}\mspace{14mu} v_{0}} > 0} \\{{{h\left( {\lambda_{0} - \pi - {2\;\beta}} \right)}\mspace{14mu}{where}\mspace{14mu} v_{0}} < 0}\end{matrix} \right.}}} & (7)\end{matrix}$

To allow a complementary ray to be obtained, the position z₁ of theradiation source has to be “shifted” to the position z₀. It musttherefore be shifted by

$\begin{matrix}{{\Delta\; ϛ} = {{z_{1} - z_{0}} = \left\{ \begin{matrix}{{{h\left( {{+ \pi} - {2\beta}} \right)} - {v_{0}\frac{2R\;\cos\;\beta}{\sqrt{u_{1}^{2} + R^{2}}}\mspace{14mu}{where}\mspace{14mu} v_{0}}} > 0} \\{{{h\left( {{- \pi} - {2\beta}} \right)} - {v_{0}\frac{2R\;\cos\;\beta}{\sqrt{u_{1}^{2} + R^{2}}}\mspace{14mu}{where}\mspace{14mu} v_{0}}} < 0}\end{matrix} \right.}} & (8)\end{matrix}$

The position of the radiation source for the complementary ray is thusthat position of the radiation source which is defined by the angularposition λ₁ and has been shifted along the axis of rotation by Δζ.

The complementary ray that starts from that radiation source position ofthe approximating ray which has been shifted by Δζ must impinge on thesurface of the cylinder at the z position from which the associateddirect ray starts. This produces the following defining equation for thecoordinate v₁ of the complementary ray:

$\begin{matrix}{{h\;\lambda_{0}} = {{h\;\lambda_{1}} + {v_{1}\frac{2R\;\cos\;\beta}{\sqrt{u_{1}^{2} + R^{2}}}}}} & (9)\end{matrix}$

By inserting equating (3), this gives

$\begin{matrix}{v_{1} = \left\{ {\begin{matrix}{{{h\left( {{2\beta} - \pi} \right)}\frac{\sqrt{u_{1}^{2} + R^{2}}}{2R\;\cos\;\beta}\mspace{14mu}{where}\mspace{14mu} v_{0}} > 0} \\{{{h\left( {{2\beta} + \pi} \right)}\frac{\sqrt{u_{1}^{2} + R^{2}}}{2R\;\cos\;\beta}\mspace{14mu}{where}\mspace{14mu} v_{0}} < 0}\end{matrix}.} \right.} & (10)\end{matrix}$

The complementary ray is therefore found from the approximating ray byshifting the radiation source along the axis of rotation by Δζ and byshifting the point of impingement of the ray on the detector by

$\begin{matrix}{{\Delta\; v} = {{v_{1} - {\overset{\sim}{v}}_{1}} = {{- {\Delta ϛ}}\;{\frac{R^{2} + u_{1}^{2}}{R^{2}}.}}}} & (11)\end{matrix}$

The u coordinate of the complementary ray is equal to the correspondingcoordinate of the approximating ray.

Once the path followed by the complementary ray complementary to adirect ray is known, the complementary measured value associated withthe complementary ray can be determined in step 109. If thecomplementary measured value was acquired, the measured value acquiredcan be used as a complementary measured value. This is not generally thecase however, because the radiation source position of the complementaryray is usually not situated on the helix 17.

From “Improved 2D rebinning of helical cone-beam CT data using John'sequation”, Defrise, Noo, Kudo, M10-74 in 2002 Nuclear Science SymposiumConference Record, guest editor Scott Metzler, 10-16 November, Norfolk,Va., USA, ISBN 0-7803-7637-4, it is known that, with the help of John'sequation, a complementary measured value that is consistent to a goodapproximation with the measured values can be calculated by adding acorrecting measured value to the approximating measured value. Thecorrecting measured value Δg(λ₁,u₁,{tilde over (v)}₁) is given by thefollowing equation:

$\begin{matrix}{{\Delta\;{g\left( {\lambda_{1},u_{1},{\overset{\sim}{v}}_{1}} \right)}} = {\frac{\Delta\; ϛ}{R}{\int_{- \infty}^{u_{1}}\mspace{7mu}{{\mathbb{d}{u^{\prime}\left( {\frac{\partial{g\left( {\lambda,u,v} \right)}}{{\partial\lambda}\;{\partial v}} + {\frac{{{Ru}^{\prime}v} - {h\left( {R^{2} + u^{\prime 2}} \right)}}{R^{2}}\frac{\partial{g\left( {\lambda,u,v} \right)}}{\partial^{2}v}}} \right)}}{_{{\lambda = \lambda_{1}},{v = {\overset{\sim}{v}}_{1}}}.}}}}} & (12)\end{matrix}$

In accordance with the invention, all or only a proportion of themeasured values acquired can be used to determine the correctingmeasured value from equation (12). In the present embodiment, themeasured values used are only ones whose associated rays start fromradiation source positions y(λ) that are located on the helix 17 in thevicinity of the radiation source position y(λ₁) of the approximatingray. What that means is that the measured values used are only oneswhose associated radiation source positions are situated in the rangey(λ) where λ₁−Δ_(λ)<λ<λ₁+Δ_(λ). In the present case, Δ_(λ) is selectedto be such that ten different radiation source positions y(λ) arecovered. In other embodiments a different number of radiation sourcepositions, and hence of measured values, can be used.

The range Δ_(λ) is selected to be at least sufficiently wide forredundant measured values to contribute to at least some complementarymeasured values. However, in the majority of cases this condition willalready have been met due to the fact that the approximating measuredvalue is itself redundant.

To determine the correcting measured value, all the measured valueswhose associated rays start from radiation source positions that arewithin the above range are derived from equation (12) by their angularpositions λ and their v coordinate, to allow a first dataset of partialderivatives to be derived from equation (12). These derivations, and thefollowing ones too, may be performed by, for example, the finitedifference method. Also, the original measured values, i.e. thenon-derived ones, whose associated rays start from radiation sourcepositions that are within the above range are twice derived as partialderivatives, for v, to allow a second dataset of derivatives to beobtained.

The measured values in the second dataset of derivatives are thenmultiplied by a weighting factor that follows a square law in u and islinear in v. In the present embodiment, this weighting factor is equalto

$\begin{matrix}{\frac{{Ruv} - {h\left( {R^{2} + u^{2}} \right)}}{R^{2}}.} & (13)\end{matrix}$

The next step is to add the measured values in the first and seconddatasets of derivatives to give a set of measured values, and thoseresulting values whose associated rays start from the angular positionλ₁ and impinge on row {tilde over (v)}₁ of the detector unit and oncolumns whose coordinates u are equal to or smaller than u₁, are addedup.

Finally, the resulting sum is multiplied by a factor that isproportional to Δζ and inversely proportional to R. The sum ispreferably multiplied by Δζ/R. The sum multiplied by the correctingfactor is the correcting measured value for the approximating measuredvalue g(λ₁,u₁,{tilde over (v)}₁), or for the measured value g(λ₀,u₀,v₀)of the direct ray.

The correcting measured value having been calculated, the complementarymeasured value g_(c)(λ₀,u₀,v₀) to the measured value g(λ₀,u₀,v₀) can bedetermined fromg _(c)(λ₀ ,u ₀,v₀)=g(λ₁ ,u ₁,{tilde over (v)}₁)+Δg(λ₁ ,u ₁,{tilde over(v)}₁)by adding the correcting measured value to the approximating measuredvalue.

Once steps 105 to 109 have been performed for all the measured valuesg(λ₀,u₀,v₀) that lie within the PI window, two sets of measured valuesexist, i.e. the dataset acquired in step 103 and the datasetcomplementary thereto.

In step 111, each measured value for which a complementary measuredvalue has been determined is replaced by a sum comprising thecomplementary measured value after it has been weighted and theassociated measured value after it has been weighted. For this purpose,the given measured value is added to the associated complementarymeasured value, each value having been multiplied by a weighting factorprior to the addition. If the complementary measured value was acquired,then it and the associated measured value are weighted by equal amounts,and in particular are each multiplied by 0.5. If the complementarymeasured value was determined from a consistency condition, such asJohn's equation for example, then it is multiplied by a smallerweighting factor than the associated measured value. In this way, themeasured value may be multiplied by 0.9 or 0.8 for example, whereas thecomplementary measured value is multiplied by 0.1 or 0.2. In the presentembodiment, the sum of a measured value/complementary measured valuepairing is equal to 1.

In the next step 113, a CT image, which in this embodiment is an imageof the distribution of absorption in the examination region, isreconstructed by an exact method from the measured values lying withinthe PI window that were determined in the previous steps, i.e. inparticular from the measured values that have been obtained by addingweighted complementary measured values and associated measured values.The individual steps in the reconstruction are shown in FIG. 7.

In other embodiments, a complementary measured value could also havebeen determined for only a proportion of the measured values that liewithin a reconstruction window. If this were the case, then what wouldbe used for the reconstruction would be, on the one hand, the measuredvalues lying within the reconstruction window that had been obtained byadding a weighted complementary measured value and a weighted measuredvalue or, on the other hand, if no complementary measured values hadbeen determined for some of the measured values lying within thereconstruction window, the measured value that was acquired in theparticular case.

To allow the exact reconstruction to be understood, the followingequation from E1 will first be cited:

$\begin{matrix}{{f(x)} = {{- \frac{1}{2\;\pi^{2}}}{\int_{I_{Pt}{(x)}}^{\;}\mspace{7mu}{{\mathbb{d}s}\frac{1}{{x - {y(\lambda)}}}{\int_{- \pi}^{\pi}\;{\frac{\mathbb{d}\gamma}{\sin\;\gamma}\frac{\partial\;}{\partial q}{D_{f}\left( {{y(q)},{{\Theta\left( {s,x,\gamma} \right)}{_{q = \lambda}.}}} \right.}}}}}}} & (15)\end{matrix}$

This equation describes an exact reconstruction of absorption byback-projection of the measured values. In it, ƒ(x) is the spatialdistribution of absorption in the examination region at point x.

The measured value Dƒ(y,Θ) can be defined by the line integral:

$\begin{matrix}{{D_{f}\left( {{y(\lambda)},\Theta} \right)} = {\int_{0}^{\infty}\mspace{7mu}{{\mathbb{d}l}\;{f\left( {y + {l\;\Theta}} \right)}}}} & (16)\end{matrix}$

Here, the unit vector Θ gives the direction of the ray belonging to themeasured value.

In step 201, the measured values are derived as partial derivatives fromequation (15) for q, i.e. for the angular position of the radiationsource at the point q=λ. It should be borne in mind in this case thatonly y depends on q and not Θ, which means that, for the derivation,account has to be taken of measured values from parallel rays in eachcase. Because parallel rays have the same cone angle, then, as shown inFIG. 8, in the case of the focus-centered detector 16 that is used inthe present case, the parallel rays 51 impinge on the same row 53 in thedetector. In this case the cone angle of a ray is the angle that the raymakes with a plane perpendicular to the axis of rotation 14. For thepartial derivation, the measured values may first be resorted. For thispurpose, measured values that belong to parallel rays, i.e. to the samerow 53 in the detector but to different angular positions λ_(a), λ_(b),λ_(c) of the radiation source, are combined in each case to form a set.The measured values in each set are derived, numerically for example bythe finite difference method, in accordance with the angular position ofthe radiation source.

The unit vector Θ depends on the κ angle γ that can be defined by meansof so-called κ planes 55 (see FIG. 9). The κ planes 55 will be explainedbelow.

To allow a κ plane 55 to be determined, a function

$\begin{matrix}{{\lambda_{1}\left( {\lambda,\lambda_{2}} \right)} = \left\{ \begin{matrix}{\frac{{m\;\lambda_{2}} + {\left( {n - m} \right)\lambda}}{n},{\lambda \leq \lambda_{2} < {\lambda + {2\;\pi}}}} \\{\frac{{m\;\lambda} + {\left( {n - m} \right)\lambda_{2}}}{n},{\lambda > \lambda_{2} > {\lambda - {2\;\pi}}}}\end{matrix} \right.} & (17)\end{matrix}$is introduced that depends on non-negative values n and m, n>m, that arewhole numbers. In the present embodiment, n is selected to be equal to 2and m to 1. Other values could however also be selected for n, m.Equation (15) would still remain exact; only the positions of the κplanes 55 would change. Also defined are the vector function

$\begin{matrix}{{u\left( {\lambda,\lambda_{2}} \right)} = \left\{ \begin{matrix}{{{\frac{\begin{matrix}{\left\lbrack {{y\left( {\lambda_{1}\left( {\lambda,\lambda_{2}} \right)} \right)} - {y(\lambda)}} \right\rbrack \times} \\\left\lbrack {{y\left( \lambda_{2} \right)} - {y(\lambda)}} \right\rbrack\end{matrix}}{\begin{matrix}{\left\lbrack {{y\left( {\lambda_{1}\left( {\lambda,\lambda_{2}} \right)} \right)} - {y(\lambda)}} \right\rbrack \times} \\\left\lbrack {{y\left( \lambda_{2} \right)} - {y(\lambda)}} \right\rbrack\end{matrix}} \cdot {sgn}}\;\left( {\lambda_{2} - \lambda} \right)},{0 < {{s_{2} - s}}}} \\{\frac{{\overset{.}{y}(\lambda)} \times {\overset{¨}{y}(\lambda)}}{{{\overset{.}{y}(\lambda)} \times {\overset{¨}{y}(\lambda)}}},{\lambda_{2} = \lambda}}\end{matrix} \right.} & (18)\end{matrix}$and the unit vector

$\begin{matrix}{{\beta\;\left( {\lambda,x} \right)} = {\frac{x - {y(\lambda)}}{{x - {y(\lambda)}}}.}} & (19)\end{matrix}$

The vector β points from the radiation source position y(λ) to positionx. To determine the κ plane, a value λ₂∈I_(PI)(x) is selected such thatyλ, y(λ₁,λ,λ₂), y(λ₂) and x lie in a plane. This plane is referred to asthe κ plane 55 and the line of intersection between the κ plane 55 andthe face 74 of the detector is referred to as the κ line 57. In FIG. 9,the detector 74 is bounded by two successive turns of the helicaltrajectory 17 and is of the same curvature as the helix 17. Thisdetector has been used here as an example to make things clearer.Corresponding lines of intersection 57 can be determined for otherdetectors, such as the focus-centered or the planar detector. In FIG. 9is shown a fan-shaped part of a κ plane. The edges of the fan meet atthe location of the radiation source. Defining the κ plane 55 in thisway is equivalent to solving the equation(x−y(λ))·u(λ,λ₂)=0, λ₂ ∈I _(PI(x))   (20)for λ₂·u is thus the normal vector of the κ plane 55. To allow thevector function Θ(λ,x,γ) to be determined, the vectore(λ,x)=β(λ,x)×u(λ,x)   (21)is defined. With the definitions for β and e, the vector functionΘ(λ,x,γ) can then be stated as follows:Θ(λ, x, γ)=cos γ·β(λ, x)+sin γ·e(λ, x)   (22)

Because the two vectors β and e are oriented perpendicularly to u, the κangle γ gives the direction of the vector Θ and hence the direction of aray within a κ plane.

κ planes and κ lines are described in detail in E1, which is herebyincorporated by reference.

In step 203, the derived measured values along κ lines are multiplied inequation (15) by a weighting factor that decreases as the sine of the κangle increases and in particular is equal to the reciprocal of the sineof the κ angle, and are added up. For this purpose, a κ line isdetermined for each point x in the examination region and for eachradiation source position λ, in which case, as explained above, a valueλ₂∈I_(PI)(x) is selected such that yλ, y(λ₁,λ,λ₂), y(λ₂) and x lie in aplane, the κ plane. The κ line is then determined as the line ofintersection between the κ plane and the face of the detector. Themultiplications by the weighting factor and the integrations oradditions may be performed by means of, for example, a Fouriertransformation.

The derived and integrated measured values can be represented by thefollowing equation:

$\begin{matrix}{{p\text{(}{y\left( {\lambda\;{\Phi\left( {\lambda,x} \right)}} \right)}} = {\int_{- \pi}^{\pi}{\frac{\mathbb{d}\;\gamma}{\sin\;\gamma\;{\partial q}}\ \frac{\partial\;}{\partial q}{D_{f}\left( {{y(q)},{\Theta\left( {\lambda,x,\gamma} \right)}} \right)}{_{q = \lambda}.}}}} & (23)\end{matrix}$

In this equation, p(y(λ), Φ(λ,x)) are the derived and integratedmeasured values and Φ(λ,x) is a unit vector that points from theradiation source position y(λ) towards point x in the examinationregion.

The integration step that is missing in equation (15), i.e. theback-projection of the measured values, can then be defined by thefollowing equation:

$\begin{matrix}{{f(x)} = {{- \frac{1}{2\;\pi^{2}}}{\int_{I_{Pt}{(x)}}^{\;}\mspace{7mu}{{\mathbb{d}\lambda}\frac{1}{{x - {y(\lambda)}}}{{p\left( {{y(\lambda)},{\Phi\left( {\lambda,x} \right)}} \right)}.}}}}} & (24)\end{matrix}$

Before the back-projection, a rebinning of the measured values can takeplace in step 207. The rebinning causes the measured values to beresorted and reinterpolated as if they had been measured with adifferent radiation source (an extended radiation source, located on apart of a helix, that is able to emit fans of rays each parallel to oneanother).

This will be explained in detail by reference to FIG. 10. In thisFigure, 17 denotes the helical trajectory from which the radiationsource passes its rays through the examination region. Reference numeral43 denotes a fan-shaped beam of rays that starts from the radiationsource position S₀ and whose rays extend along paths in a planecontaining the axis of rotation 14. The conical beam of rays that isemitted by the radiation source at the position S₀ can be thought of ascomposed of a plurality of plane fans of rays that are situated inplanes parallel to the axis of rotation 14 and that intersect at theradiation source position S₀. Of these fans of rays FIG. 10 shows onlyone, namely fan of rays 43.

Also shown in FIG. 10 are further fans of rays 41, 42 and 44, 45 thatare parallel to fan of rays 43 and that lie in planes parallel to oneanother and to the axis of rotation 14. The associated radiation sourcepositions S⁻², S⁻¹ and S₁, S₂ are occupied by the radiation source Srespectively before and after it has reached radiation source positionS₀. All the rays in the fans of rays 41 to 45 have the same angle ofprojection. The fans of rays 41 to 45 define a beam of rays of atent-like shape.

The measured values obtained after the rebinning are then used toreconstruct the distribution of absorption in the examination region bymeans of a back-projection, carried out with equation (23) in thepresent embodiment.

For this purpose, in step 209 a voxel V(x) within a presettable region(the field of view (FOV)) in the examination region, and a PI intervalI_(PI)(x) for this voxel, are determined. Then, in step 211, an angularposition λ within the interval I_(PI)(x) is preset. In step 213, a checkis made to see whether a measured value whose ray passes through thecenter of the voxel V(x) is available for the angular position λ. If nosuch ray can be found, then it is found at what point a central raywould have impinged on the face of the detector. The associated measuredvalue is then calculated by interpolation from the measured values fromadjacent rays. The measured value that can be assigned to the raypassing though the voxel, or the measured value obtained byinterpolation, as the case may be, is multiplied in step 214 by aweighting factor that becomes smaller as the distance between theradiation source y(λ) and the point x to be reconstructed in theexamination region increases. In the present embodiment, this weightingfactor is, as shown in equation (24), equal to 1/|x−y(λ)|. In step 215,the weighted measured value is summed to give the voxel V(x). In step217 a check is made to see whether all the angular positions λ in theinterval I_(PI)(x) have been looked at. If they have not, the flow chartbranches back to step 211. Otherwise, a check is made in step 219 to seewhether all the voxels V(x) in the FOV have been covered. If they havenot, the sequence continues with step 209. If on the other hand all thevoxels V(x) in the FOV have been covered, then the absorption in thewhole of the FOV, and hence the CT image, has been determined, and theexact reconstruction of the CT image comes to an end in step 221.

As has already been mentioned above, in other embodiments the radiationsource may also move along a circular trajectory relative to theexamination region. If, in a case of this nature, the measured valuesused to reconstruct a CT image are only ones that lie within areconstruction window, then in this case too complementary measuredvalues can be determined for at least some measured values, inparticular by means of John's equation. As was described above for thehelical trajectory, a sum that replaces the measured value in theacquired dataset is formed from a measured value/complementary measuredvalue pairing. Finally, a CT image can be reconstructed by known methodsof reconstruction from the data that has been obtained in this way.

A computer tomography method in which the radiation source movesrelative to the examination region along a circular trajectory and themeasured values used are solely ones that lie within a reconstructionwindow is known from, for example, “A fast and efficient method forsequential cone-beam CT”, Koehler, Proksa, Grass, Medical Physics, vol.28, no. 11, pp. 2318-2327.

LIST OF REFERENCE NUMERALS

-   α_(max) Included angle-   λ, λ₁, λ₂, λ_(a), λ_(b), λ_(c) Angular positions of the radiation    source on the helical trajectory-   Δζ Shift in the z direction-   Δv Shift in the v direction-   x Point in the examination region-   I_(PI)(x) Section of helix-   S Radiation source-   S⁻², S⁻¹, S₀, S₁, S₂ Positions of radiation source-   1 Gantry-   2, 5 Motors-   3 Collimator arrangement-   4 Beam of rays-   7 Control unit-   10 Image processing computer-   11 Monitor-   13 Examination region-   14 Axis of rotation-   16 Detector unit-   17 Helical trajectory-   21, 23 PI lines-   25 PI window-   31 Approximating ray-   33 Direct ray-   35 Surface of cylinder-   37 PI straight line-   41, 42, 43, 44, 45 Mutually parallel fans of rays-   51 Parallel rays-   53 Row on detector-   55 κ plane-   57 κ line-   60 Planar detector-   70 Beam of rays-   74 Face of detector

1. A computer tomography method having the following steps: a)generation, by a radiation source, of a conical beam of rays that passesthrough an examination region and an object situated therein, b)production of a relative movement between the radiation source on theone hand and the examination region on the other hand, which movementcomprises at least a rotary movement about an axis of rotation and is inparticular in the form of a helix or circle, c) acquisition, by adetector unit and during the relative movement, of measured values thatdepend on the intensity in the beam of rays on the farther side of theexamination region, d) determination, with the help of redundantmeasured values, of a complementary measured value for each of at leastsome of the measured values that were acquired in step c) and that liewithin a reconstruction window, the rays associated with the givenmeasured value and the complementary measured value belonging to itbeing oriented in opposite directions to one another, e) replacement ofeach measured value for which a complementary measured value wasdetermined in step d) by a sum comprising the measured value, havingbeen weighted, and the complementary measured value, having beenweighted, f) reconstruction of a CT image of the examination region fromthe measured values lying within the reconstruction window.
 2. Acomputer tomography method as claimed in claim 1, wherein, if a ray thatis associated with a complementary measured value from step d) followsthe same path as a ray that is associated with a measured value that wasacquired in step c), the complementary measured value is set to be equalto this measured value, and in that, if a ray that is associated with acomplementary measured value from step d) does not follow the same pathas one of the rays that are associated with the measured values thatwere acquired in step c), the complementary measured value in questionis determined with the help of John's equation.
 3. A computer tomographymethod as claimed in claim 1, wherein, prior to the addition in step e),the complementary measured value and the associated measured value areeach multiplied by a weighting factor, the weighting factors forcomplementary measured value and the measured value belonging theretobeing equal if the ray that is associated with the complementarymeasured value from step d) follows the same path as one of the raysthat are associated with the measured values that were acquired in stepc), and the weighting factor for a measured value being greater than theweighting factor for an associated complementary measured value if theray that is associated with the complementary measured value from stepd) does not follow the same path as one of the rays that are associatedwith the measured values that were acquired in step c).
 4. A computertomography method as claimed in claim 1, wherein, in step c) therelative movement is in the form of a helix and in that thereconstruction of a CT image in step f) comprises the following steps:partial derivation of measured values with which parallel rays havingdifferent radiation source positions are associated, for an angularposition of the radiation source on the helix that is associated withthe given measured with value, filtering of the derived measured valuesalong κ lines, reconstruction of the CT image by back-projection ofthose filtered measured values that lie within a PI window.
 5. Acomputer tomography method is claimed in claim 4, wherein the filteringof a measured value comprises the following steps: determination of a κline for the measured value, multiplication of those measured value thatare situated on the κ line by a weighting factor that increases with thereciprocal of the sine of the κ angle-and in particular is equal to thisreciprocal, adding up of the weighted measured values lying on the κline, the resulting sum being the filtered measured value.
 6. A computertomograph having a radiation source for generating a conical beam ofrays that passes through an examination region and an object situatedtherein, a drive arrangement to enable an object contained in theexamination region and the radiation source to be caused to rotaterelative to one another about an axis of rotation and to be displacedparallel to the axis of rotation, a detector unit coupled to theradiation source, that has a detector surface, for acquired measuredvalues, a reconstructing unit for reconstructing a CT image within theexamination region from the measured values acquired by the detectorunit a control unit for controlling the radiation source, the detectorunit the drive arrangement and the reconstructing unit in the followingsteps: a) generation by the radiation source of a conical beam of raysthat passes through the examination region and the object situatedtherein, b) production of a relative movement between the radiationsource on the one hand and the examination region on the other hand,which movement comprises at least a rotary movement about the, axis ofrotation and is in particular in the form of a helix or circle, c)acquisition, by the detector unit and during the relative movement, ofmeasured values that depend on the intensity in the beam of rays on thefarther side of the examination region, d) determination, with the helpof redundant measured values, of a complementary measured value for eachof at least some of the measured values that were acquired in step c)and that lie within a reconstruction window, the rays associated withthe given measured value and the complementary measured value belongingto it being oriented in opposite directions, to one another, e)replacement of each measured value for which a complementary measuredvalue was, determined in step d) by a sum comprising the measured value,having been weighted, and the complementary measured value, having beenweighted, f) reconstruction of a CT image of the examination region fromthe measured values lying within the reconstruction window.
 7. Acomputer readable medium encoded with a computer program for a controlunit for controlling a radiation source, a detector unit, a drivearrangement and a reconstructing unit of a computer tomograph to performin the following sequence: a) generation,by the radiation source (S), ofa conical beam of rays that passes through the examination region andthe object situated therein, b) production of a relative movementbetween the radiation source on the one hand and the examination regionon the other hand, which movement comprises at least a rotary movementabout the axis of rotation and is in particular in the form of a helixor circle, c) acquisition, by the detector unit and during the relativemovement, of measured values that depend on the intensity in the beam ofrays on the farther side of the examination region, d) determinationwith help of redundant measured values, of a complementary measuredvalue for each of at least some of the measured values that wereacquired in step c) and that lie within a reconstruction window, therays associated with the given measured value and the complementarymeasured value belonging to it being oriented in opposite directions toone another, e) replacement of each measured value for which acomplementary measured value was determined in step d) by a sumcomprising the measured value, having been weighted, and thecomplementary measured value, having been weighted, f) reconstruction ofa CT image of the examination region from the measured values lyingwithin the reconstruction window.